Comparison of substructured non-overlapping domain decomposition and overlapping additive Schwarz methods for large-scale Helmholtz problems with multiple sources

TL;DR

This paper compares non-overlapping substructured domain decomposition methods with overlapping additive Schwarz methods for large-scale 3D Helmholtz problems, demonstrating that non-overlapping approaches can outperform overlapping ones when properly tuned.

Contribution

It provides a comparative analysis of non-overlapping and overlapping domain decomposition methods for Helmholtz problems, highlighting conditions under which non-overlapping methods are more efficient.

Findings

Non-overlapping methods can significantly outperform overlapping methods when properly tuned.

The study uses realistic geophysical test cases to validate performance differences.

Proper parameter tuning is crucial for the effectiveness of non-overlapping domain decomposition methods.

Abstract

Solving large-scale Helmholtz problems discretized with high-order finite elements is notoriously difficult, especially in 3D where direct factorization of the system matrix is very expensive and memory demanding, and robust convergence of iterative methods is difficult to obtain. Domain decomposition methods (DDM) constitute one of the most promising strategy so far, by combining direct and iterative approaches: using direct solvers on overlapping or non-overlapping subdomains, as a preconditioner for a Krylov subspace method on the original Helmholtz system or as an iterative solver on a substructured problem involving field values or Lagrange multipliers on the interfaces between the subdomains. In this work we compare the computational performance of non-overlapping substructured DDM and Optimized Restricted Additive Schwarz (ORAS) preconditioners for solving large-scale Helmholtz problems with multiple sources, as is encountered, e.g., in frequency-domain Full Waveform Inversion. We show on a realistic geophysical test-case that, when appropriately tuned, the non-overlapping methods can reduce the convergence gap sufficiently to significantly outperform the overlapping methods.